lect1_06jan17

# lect1_06jan17 - Imbens Lecture Notes 1 ARE213 Spring'06...

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Imbens, Lecture Notes 1, ARE213 Spring ’06 1 ARE213 Econometrics Spring 2006 UC Berkeley Department of Agricultural and Resource Economics Ordinary Least Squares I: Estimation, Inference and Predicting Outcomes (W 4.2.1-4) Let us review the basics of the linear model. We have N units (individuals, Frms, or other economic agents) drawn randomly from a large population. On each unit we observe on outcome Y i for unit i , and a K -dimensional column vector of explanatory variables X i = ( X i 1 ,X i 2 ,...,X iK ) ± (where typically the Frst covariate is a constant, X i 1 = 1 for all i = 1 ,...,N .) We are interested in explaining the distribution of Y i in terms of the explanatory variables X i using a linear model: Y i = β ± X i + ε i . (1) In this equation β is a K -dimensional column vector. In matrix notation, Y = X β + ε, where Y is an N -dimensional column vector, and X is an N × K dimensional matrix with i th row equal to X ± i . Avoiding vector and matrix notation completely: Y i = β 1 · X i 1 + ... + β K · X iK + ε i = K ± k =1 β k · X ik + ε i . We assume that the residuals ε i are independent of the covariates or regressors, and normally distributed with mean zero and variance σ 2 : Assumption 1 ε i | X i ∼N (0 2 ) . We can weaken this considerably. ±irst, we could relax normality and only assume indepen- dence:

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Imbens, Lecture Notes 1, ARE213 Spring ’06 2 Assumption 2 ε i X i , combined with the normalization that E [ ε i ] = 0. We can even weaken this assumption further by requiring only mean-independence Assumption 3 E [ ε i | X i ]=0 , or even further, requiring only zero correlation: Assumption 4 E [ ε i · X i ]=0 . We will also assume that the observations are drawn randomly from some population. We can also do most of the analysis by assuming that the covariates are Fxed, but this complicates matters for some results, and it does not help very much. See the discussion on Fxed versus random covariates in Wooldridge (page 9) Assumption 5 The pairs ( X i ,Y i ) are independent draws from some distribution, with the Frst two moments of X i Fnite. The (ordinary) least squares estimator for β solves min β N ± i =1 ( Y i - β ± X i ) 2 . This leads to ˆ β =( X ± X ) - 1 ( X ± Y ) . The (exact) distribution of the ols estimator under the normality assumption in Assumption 1is ˆ β ∼N ² β,σ 2 · ( X ± X ) - 1 ³ .
Imbens, Lecture Notes 1, ARE213 Spring ’06 3 Without the normality of the ε it is diﬃcult to derive the exact distribution of ˆ β . However, under the independence Assumption 2 and a second moment condition on ε (variance Fnite and equal to σ 2 ), we can establish asymptotic normality: N ( ˆ β - β ) d -→ N ( 0 2 · E [ XX ± ] - 1 ) .

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lect1_06jan17 - Imbens Lecture Notes 1 ARE213 Spring'06...

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